Mechanics of Viscous Fingering in Miscible Systems

Abstract

A simplified theory of viscous fingering in miscible systems has been developed. It predicts the correct functional relationship between pertinent variables and permits the calculation of the order of magnitude of fingering behavior for simple systems, such as linear or radial. The theory is based on the following four observations:cross-flow takes place only near the ends of fingers;the relative finger width is about 0.5;fingers can be suppressed by transverse dispersion, the suppression being quantitatively described by a critical ratio of dispersion times; andthe widths of extending fingers are close to the minimum size finger that can grow at any stage of displacement. Fingering is studied in two-dimensional linear and diverging radial systems, both theoretically and experimentally. For linear systems, the length of the fingered region is proportional to mean displacement; the finger width is proportional to square root of mean displacement; and there is a small initial region void of fingers because of suppression by longitudinal dispersion. For the radial system, two limiting conditions are recognized. If the mean displacement is small compared with the wellbore radius, the length of the fingered region is proportional to the mean displacement. The width is proportional to the square root of mean displacement. If the mean displacement is large compared with wellbore radius, length of the fingered region is proportional to mean displacement, but the number of fingers approaches a constant value. Also, in the radial case there is a small initial region void of fingers because of longitudinal dispersion. Introduction The behavior of viscous fingers in miscible systems has been of interest to the oil industry for many years. Previous studies have clearly shown the existence and nature of fingers in small models,1,2,3,4 Engineering techniques for extrapolating to reservoir situations have been proposed.5 Still, because of the lack of a real understanding of the mechanics of fingering, there remains uncertainty and disagreement as to the best way to scale or calculate fingering behavior in the reservoir. In this paper we discuss a study of the mechanics of fingering in miscible systems (i.e., why do fingers behave as they do?). A simplified theory is developed which we believe willpredict the correct functional relationship between pertinent variables,permit us to calculate the order of magnitude of fingering behavior for simple systems such as linear and radial, andgive further insight into the problem of scaling or otherwise extending the results to more complicated reservoir conditions. The paper includes the following sections:brief summary of the mixing behavior of miscible fluids in linear and radial systems;four fundamental observations of fluid flow (under fingering conditions);based on these four observations, a simplified theory of fingering in linear and radial systems is developed; andthe theoretical equations are compared with experimental fingering data measured in laboratory models. A Review of Diffusion and Dispersion As will be shown later, the behavior of viscous fingers in miscible systems is controlled in large degree by the mixing between the two fluids. A quantitative description of fingers will first require a quantitative description of mixing. Fortunately, much work has been done to clarify this subject; a fairly comprehensive review has been presented in a previous paper.7 For present purposes let usbriefly summarize the quantitative description of diffusion and dispersion (both longitudinal and transverse) within a differential element;present a simplification of the "width of mixed zone"; anddescribe the effect of geometry on width of mixed zone.

Simulation of multiphase flows in porous media

“…The effects of dispersion in various flow processes have been discussed extensively in the literature [46,57,58,75,76,80]. Russell and Wheeler [67] and Young [83] have given excellent surveys of the influence of dispersion and attempts to incorporate it in present reservoir simulators.…”

“…This phenomenon, termed viscous fingering, is well known [4,12,39,40,44,47,[51][52][53][54][55][56][57][58][59][60][61][62][63][64][68][69][70][71][72]78,79,81,82] and is a serious problem in hydrocarbon recovery. This problem and different techniques for understanding and modeling it were surveyed in [26].…”

Promise and Problems of Miscible-Flood Enhanced Oil Recovery

“…Adverse mobility ratios produce "viscous fingering," a displacement phenomenon that has been known and studied both experimentally and theoretically for many years (9)(10)(11)(12)(13)(14). This phenomenon was a principal reason for the failure of many of the liquified petroleum gas (LPG) floods of the 1950's and early 1960's (15).…”

An extended theory to predict the onset of viscous instabilities for miscible displacements in porous media

“…These two problems arise as a result of the phenomenon of viscous fingering which comes about due to hydrodynamic instability of the displacement front (Lacey et aL, 1958;Blackwell et al, 1959;Habermann, 1960;Perkins et al, 1965). This can prevent the recovery of significant amounts of oil.…”